A 1-year European call option on a stock with a continuous dividend rate of 2%

Su = price in the up state = 60; Sd = price in the down stgatge = 40; S0 = Current price = 50; Strike price, K = 55

Continuous div rate, ? = 2%; Risk free rate = r = 5%; Time to maturity, t = 1 year

up factor, u = Su/S0 = 60/50 = 1.2

down factor, d = Sd/S0 = 40/50 = 0.8

Risk neutral probability of an up state = p = [e^{(r - ?) x t} - d] / (u - d) = [e^{(5% - 2%) x 1} - 0.8] / (1.2 - 0.8) = 0.5761

Payoff of the call option in up state = Cu = max (Su - K, 0) = max (60 - 50, 0) = 10

Payoff of the call option in down state = Cd = max (Sd - K, 0) = max (40 - 50, 0) = 0

Hence, the option premium = [p x Cu + (1 - p) x Cd] x e^{-r x t} = [0.5761 x 10 + (1 - 0.5761) x 0] x e^{-5% x 1} = 5.48